I know that with the class of "high-resolution" schemes, we may get well-accepted solution with only 500 cells. But if I have to stick to compact schemes, in which the slope reconstruction & limiting can only involve the face-neighboring cells of a cell, what could be the "best" choices so far?

I started practicing with Green-Gauss or least-squares slope reconstruction plus min-max slope limiter. The result looks not so good even on a grid of 5000 cells.

I know that with the class of "high-resolution" schemes, we may get well-accepted solution with only 500 cells. But if I have to stick to compact schemes, in which the slope reconstruction & limiting can only involve the face-neighboring cells of a cell, what could be the "best" choices so far?

I started practicing with Green-Gauss or least-squares slope reconstruction plus min-max slope limiter. The result looks not so good even on a grid of 5000 cells.

I appreciate any more comments that will be shared here.

have you checked the methods for the W-C proposed in the book of LeVeque?

I know that with the class of "high-resolution" schemes, we may get well-accepted solution with only 500 cells. But if I have to stick to compact schemes, in which the slope reconstruction & limiting can only involve the face-neighboring cells of a cell, what could be the "best" choices so far?

I started practicing with Green-Gauss or least-squares slope reconstruction plus min-max slope limiter. The result looks not so good even on a grid of 5000 cells.

I appreciate any more comments that will be shared here.

What limiter did you use? I am focusing this problem with high order method. FV's results with minmod limiter looks fine to me. You can discuss with me if you are still interested.

What limiter did you use? I am focusing this problem with high order method. FV's results with minmod limiter looks fine to me. You can discuss with me if you are still interested.

Thanks for your reply. When I posted this thread initially, I used the conservative variables (e.g., rho, rhou, rhoe) in my limiting process for 1D. The limiters I tested include minmod, superbee, and MC. Those are typical limiters in textbook, as we all know. However, I still observed obvious over-/under-shooting in some part of the solution. Later on, I used primitive variables (p, u, T) in the minmod limiter, which resulted essentially non-oscillating results. I assume that [Tip No.1] is we should choose the primitive-variable based limiters over the conservative-variable limiters for those problems. Correct me if my understanding or experience does not make sense to you.

Second, to solve Sedov-blast-wave problem using 2D triangular or 3D tetrahedral grids, I've found that the 2nd-order FVM (with linear gradient reconstruction + min-max slope limiter) has the difficulty to survive the initial steps. Later on I used a compact-stencil WENO limiter, with which my code survived the initial extreme discontinuities. Thus I am wondering what approach have you tried to make a 2nd-order FVM robust enough to deal with a wide range of shock/blast/discontinuity problems in multi-dimensional unstructured grids.

I am not interesting in structured meshes in 1D/2D/3D, as I know the limiting on the structured meshes are much better resolved.

Thanks for your reply. When I posted this thread initially, I used the conservative variables (e.g., rho, rhou, rhoe) in my limiting process for 1D. The limiters I tested include minmod, superbee, and MC. Those are typical limiters in textbook, as we all know. However, I still observed obvious over-/under-shooting in some part of the solution. Later on, I used primitive variables (p, u, T) in the minmod limiter, which resulted essentially non-oscillating results. I assume that [Tip No.1] is we should choose the primitive-variable based limiters over the conservative-variable limiters for those problems. Correct me if my understanding or experience does not make sense to you.

Second, to solve Sedov-blast-wave problem using 2D triangular or 3D tetrahedral grids, I've found that the 2nd-order FVM (with linear gradient reconstruction + min-max slope limiter) has the difficulty to survive the initial steps. Later on I used a compact-stencil WENO limiter, with which my code survived the initial extreme discontinuities. Thus I am wondering what approach have you tried to make a 2nd-order FVM robust enough to deal with a wide range of shock/blast/discontinuity problems in multi-dimensional unstructured grids.

I am not interesting in structured meshes in 1D/2D/3D, as I know the limiting on the structured meshes are much better resolved.

It can be applied to unstructured grids. If you are interested in high order family, maybe it will help you. I would like to know how you apply limiter strategy into boundary, say 1D case.

I've tried WENO limiters and still been testing. The code blows. I upload my subroutine for trouble cell reconstruction. Its idea is from the paper:
A simple weighted essentially non-oscillatory limiter for the CPR framework.

It can be applied to unstructured grids. If you are interested in high order family, maybe it will help you. I would like to know how you apply limiter strategy into boundary, say 1D case.

I've tried WENO limiters and still been testing. The code blows. I upload my subroutine for trouble cell reconstruction. Its idea is from the paper:
A simple weighted essentially non-oscillatory limiter for the CPR framework.

Hope you could share your experiences and wisdom.

Thanks for sharing your experience.

o-- The compact WENO scheme I had used is based on the following paper. I believe there is no special treatment of the WENO limiter for a boundary cell in 1D/2D/3D. Though it has an oscillation indicator, yet it applies the very same algorithm to all the cells in the domain. Maybe I did not fully catch your described "blow-up" issue, yet the only experience I could probably share about this is that you may want to play with some parameter in the WENO scheme to make it more diffusive.

o-- I am also interested in the paper you introduced, and read part of the paper as for a quick overview. One concern I have is that the paper does not test the blast wave problems (which are of course much more challenging), e.g., the Woodward-Collela blast wave test, and the Sedov blast wave test. Maybe some later papers based on this limiter have done such verification. Did you happen to know the limitation of this limiter?

1. I haven't test the minmod limiter（Yang, Michael, and Zhi-Jian Wang） for blast wave, yet could handle that as this type limiter is designed for SD, a high resolution method. Go through this paper, you may find what you want.
Park J S, Chang T K, Kim C. Higher-order multi-dimensional limiting strategy for correction procedure via reconstruction[C]//52nd Aerospace Science Meeting. 2014: 2014-0772.

2. ＂I assume that [Tip No.1] is we should choose the primitive-variable based limiters over the conservative-variable limiters for those problems．＂
I think the following paper by C-Wang Shu could give you some proofs, A simple weighted essentially non-oscillatory limiter for the CPR framework.
He found that schemes limiting Riemann Variables are more robust than those limiting conservative variables especially for high-order scheme.(order>=3)

I used to the very WENO limiter in Shu's paper to solve a 1D nozzle shock problem. The exact solution of denstiy is in the attachment. I have no idea what kind of shock it is: strong, or weak? " Blows up" means when numerical solution reaches the extremum, the code diverges very quickly.But when numerical hasn't reach that far from the initial conditions, the code could successfully reconstruct the trouble cell. I don' t know where goes wrong and appreciate any comments. I would be very grateful.

3. I searched your name on google and happened to find that you interested key words are: DG, WENO, Compact Schemes, Unstructured Grids, etc. There's much similarity with my research field: CPR scheme.

If you want to discuss with me please send me your email address, if you don't mind. Hope we can keep contact.

1. I haven't test the minmod limiter（Yang, Michael, and Zhi-Jian Wang） for blast wave, yet could handle that as this type limiter is designed for SD, a high resolution method. Go through this paper, you may find what you want.
Park J S, Chang T K, Kim C. Higher-order multi-dimensional limiting strategy for correction procedure via reconstruction[C]//52nd Aerospace Science Meeting. 2014: 2014-0772.

2. ＂I assume that [Tip No.1] is we should choose the primitive-variable based limiters over the conservative-variable limiters for those problems．＂
I think the following paper by C-Wang Shu could give you some proofs, A simple weighted essentially non-oscillatory limiter for the CPR framework.
He found that schemes limiting Riemann Variables are more robust than those limiting conservative variables especially for high-order scheme.(order>=3)

I used to the very WENO limiter in Shu's paper to solve a 1D nozzle shock problem. The exact solution of denstiy is in the attachment. I have no idea what kind of shock it is: strong, or weak? " Blows up" means when numerical solution reaches the extremum, the code diverges very quickly.But when numerical hasn't reach that far from the initial conditions, the code could successfully reconstruct the trouble cell. I don' t know where goes wrong and appreciate any comments. I would be very grateful.

3. I searched your name on google and happened to find that you interested key words are: DG, WENO, Compact Schemes, Unstructured Grids, etc. There's much similarity with my research field: CPR scheme.

If you want to discuss with me please send me your email address, if you don't mind. Hope we can keep contact.

o-- I read your MS doc file. I would say the shock in your nozzle problem is sort of strong, but not so extreme like those problems I deal with.

o-- By the way, my main interests are 2nd-/3rd-order DG and stabilization methods that are robust enough to be implemented in engineering codes. I do not purse the order higher than 3rd, as those higher-oder methods are still far away from being appreciated and accepted in production-level code development in industry.

o-- I know CPR a little bit, as I often came across ZJ Wang's and his students's presentations in previous AIAA meetings.